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Tr. Mat. Inst. Steklova, 1998, Volume 222, Pages 3–191 (Mi tm650)  

This article is cited in 41 scientific papers (total in 42 papers)

Asymptotic Methods of Investigation of Periodic Solutions of Nonlinear Hyperbolic Equations

A. Yu. Kolesova, E. F. Mishchenkob, N. Kh. Rozovc

a P. G. Demidov Yaroslavl State University
b Steklov Mathematical Institute, Russian Academy of Sciences
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The work deals with the asymptotic theory of time periodic solutions of hyperbolic type partial differential equations which simulate oscillation processes in self-excited oscillators with distributed parameters. Peculiarities of the dynamics of the equations in question, including gradient catastrophes, are established and the part played by resonance as a source of relaxation oscillation is revealed. The bufferness phenomenon observed in physical systems is theoretically justified.
The work is intended for researchers, higher school teachers, post-graduates who deal with differential equations and their applications, and for specialists who are interested in mathematical, physical and engeneering problems of the oscillation theory.

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English version:
Proceedings of the Steklov Institute of Mathematics, 1998, 222, 1–189

Bibliographic databases:
UDC: 517
Received in February 1998

Citation: A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Asymptotic Methods of Investigation of Periodic Solutions of Nonlinear Hyperbolic Equations”, Tr. Mat. Inst. Steklova, 222, Nauka, MAIK Nauka, M., 1998, 3–191; Proc. Steklov Inst. Math., 222 (1998), 1–189

Citation in format AMSBIB
\by A.~Yu.~Kolesov, E.~F.~Mishchenko, N.~Kh.~Rozov
\paper Asymptotic Methods of Investigation of Periodic Solutions of Nonlinear Hyperbolic Equations
\serial Tr. Mat. Inst. Steklova
\yr 1998
\vol 222
\pages 3--191
\publ Nauka, MAIK Nauka
\publaddr M.
\jour Proc. Steklov Inst. Math.
\yr 1998
\vol 222
\pages 1--189

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    This publication is cited in the following articles:
    1. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “The buffer property in resonance systems of non-linear hyperbolic equations”, Russian Math. Surveys, 55:2 (2000), 297–321  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. A. Yu. Kolesov, N. Kh. Rozov, “Parametric excitation of high-mode oscillations for a non-linear telegraph equation”, Sb. Math., 191:8 (2000), 1147–1169  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. A. Yu. Kolesov, N. Kh. Rozov, “Characteristic features of the dynamics of the Ginzburg–Landau equation in a plane domain”, Theoret. and Math. Phys., 125:2 (2000), 1476–1488  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Kolesov A.Y., Rozov N.K., “Autooscillations in the Vitt system with a resonance spectrum of the proper frequencies”, Doklady Akademii Nauk, 370:1 (2000), 15–18  mathnet  mathscinet  zmath  isi
    5. A. Yu. Kolesov, N. Kh. Rozov, “The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity”, Math. Notes, 69:6 (2001), 790–798  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. A. Yu. Kolesov, N. Kh. Rozov, “Impact of quadratic non-linearity on the dynamics of periodic solutions of a wave equation”, Sb. Math., 193:1 (2002), 93–118  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. A. Yu. Kolesov, N. Kh. Rozov, “Multifrequency parametric resonance in a non-linear wave equation”, Izv. Math., 66:6 (2002), 1131–1145  mathnet  crossref  crossref  mathscinet  zmath
    8. A. Yu. Kolesov, N. Kh. Rozov, “The existence of countably many stable cycles for a generalized cubic Schrödinger equation in a planar domain”, Izv. Math., 67:6 (2003), 1213–1242  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Asanova A.T., Dzhumabaev D.S., “Correct solvability of a nonlocal boundary value problem for systems of hyperbolic equations”, Doklady Mathematics, 68:1 (2003), 46–48  mathscinet  zmath  isi
    10. Kolesov A.Y., Rozov N.K., “The buffer phenomenon in combustion theory”, Doklady Mathematics, 69:3 (2004), 469–472  mathscinet  isi
    11. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Buffer Phenomenon in Nonlinear Physics”, Proc. Steklov Inst. Math., 250 (2005), 102–168  mathnet  mathscinet  zmath
    12. Asanova A.T., Dzhumabaev D.S., “Well–posed solvability of nonlocal boundary value problems for systems of hyperbolic equations”, Differential Equations, 41:3 (2005), 352–363  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    13. Glyzin S.D., Kolesov A.Y., Rozov N.K., “Chaotic buffering property in chains of coupled oscillators”, Differential Equations, 41:1 (2005), 41–49  mathnet  crossref  mathscinet  zmath  isi  scopus  scopus
    14. A. T. Asanova, “On the unique solvability of a family of two-point boundary-value problems for systems of ordinary differential equations”, J. Math. Sci., 150:5 (2008), 2302–2316  mathnet  crossref  mathscinet  zmath
    15. A. Yu. Kolesov, N. Kh. Rozov, “Smoothing the discontinuous oscillations in the mathematical model of an oscillator with distributed parameters”, Izv. Math., 70:6 (2006), 1201–1224  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. A. Yu. Kolesov, N. Kh. Rozov, “The buffer property in a non-classical hyperbolic boundary-value problem from radiophysics”, Sb. Math., 197:6 (2006), 853–885  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    17. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Buffer phenomenon in systems close to two-dimensional Hamiltonian ones”, Proc. Steklov Inst. Math. (Suppl.), 253, suppl. 1 (2006), S117–S150  mathnet  crossref  mathscinet  zmath  elib
    18. Asanova A.T., “A nonlocal boundary value problem for systems of quasilinear hyperbolic equations”, Doklady Mathematics, 74:3 (2006), 787–790  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    19. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Buffer phenomenon in systems with one and a half degrees of freedom”, Comput. Math. Math. Phys., 46:9 (2006), 1503–1514  mathnet  crossref  mathscinet  elib  elib
    20. A. Yu. Kolesov, N. Kh. Rozov, V. A. Sadovnichii, “The problem of birth of autowaves in parabolic systems with small diffusion”, Sb. Math., 198:11 (2007), 1599–1636  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    21. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “New Methods for Proving the Existence and Stability of Periodic Solutions in Singularly Perturbed Delay Systems”, Proc. Steklov Inst. Math., 259 (2007), 101–127  mathnet  crossref  mathscinet  zmath  elib  elib
    22. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “The Buffer Phenomenon in One-Dimensional Piecewise Linear Mapping in Radiophysics”, Math. Notes, 81:4 (2007), 449–455  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    23. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Resonance Dynamics of Nonlinear Flutter Systems”, Proc. Steklov Inst. Math., 261 (2008), 149–170  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    24. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Extremal dynamics of the generalized Hutchinson equation”, Comput. Math. Math. Phys., 49:1 (2009), 71–83  mathnet  crossref  mathscinet  isi  elib  elib
    25. Dunmyre J.R., Rubin J.E., “Optimal Intrinsic Dynamics for Bursting in a Three–Cell Network”, SIAM Journal on Applied Dynamical Systems, 9:1 (2010), 154–187  crossref  mathscinet  zmath  adsnasa  isi
    26. N. Kh. Rozov, “Fenomen bufernosti v matematicheskikh modelyakh estestvoznaniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 3, 58–63  mathnet  elib
    27. A. Yu. Kolesov, E. F. Mischenko, N. Kh. Rozov, “Mnogochastotnye avtokolebaniya v dvukhmernykh reshetkakh svyazannykh ostsillyatorov”, Tr. IMM UrO RAN, 16, no. 5, 2010, 82–94  mathnet  elib
    28. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Multifrequency self-oscillations in two-dimensional lattices of coupled oscillators”, Izv. Math., 75:3 (2011), 539–567  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    29. D. V. Anosov, S. M. Aseev, R. V. Gamkrelidze, S. P. Konovalov, M. S. Nikol'skii, N. Kh. Rozov, “Evgenii Frolovich Mishchenko (on the 90th anniversary of his birth)”, Russian Math. Surveys, 67:2 (2012), 385–402  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    30. Glyzin S.D., Kolesov A.Yu., Rozov N.Kh., “Buffer Phenomenon in Neurodynamics”, Dokl. Math., 85:2 (2012), 297–300  crossref  mathscinet  zmath  isi  elib  elib  scopus
    31. A. Yu. Kolesov, N. Kh. Rozov, “Invariant tori for a class of nonlinear evolution equations”, Sb. Math., 204:6 (2013), 824–868  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    32. M. M. Preobrazhenskaya, “Primenenie metoda kvazinormalnykh form k matematicheskoi modeli otdelnogo neirona”, Model. i analiz inform. sistem, 21:5 (2014), 38–48  mathnet
    33. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Self-excited relaxation oscillations in networks of impulse neurons”, Russian Math. Surveys, 70:3 (2015), 383–452  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    34. Di H., Shang Ya., “Global existence and nonexistence of solutions for the nonlinear pseudo-parabolic equation with a memory term”, Math. Meth. Appl. Sci., 38:17 (2015), 3923–3936  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    35. A. T. Asanova, “Criteria of unique solvability of nonlocal boundary-value problem for systems of hyperbolic equations with mixed derivatives”, Russian Math. (Iz. VUZ), 60:5 (2016), 1–17  mathnet  crossref  isi
    36. Glyzin S.D. Kolesov A.Yu. Rozov N.Kh., “Self-Sustained Relaxation Oscillations in Time-Delay Neural Systems”, Murphys-Hsfs-2014: 7Th International Workshop on Multi-Rate Processes & Hysteresis (Murphys) & the 2Nd International Workshop on Hysteresis and Slow-Fast Systems (Hsfs), Journal of Physics Conference Series, 727, ed. Klein O. Dimian M. Gurevich P. Knees D. Rachinskii D. Tikhomirov S., IOP Publishing Ltd, 2016, UNSP 012004  crossref  isi  scopus  scopus
    37. Di H., Shang Ya., Peng X., “Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term”, Math. Nachr., 289:11-12 (2016), 1408–1432  crossref  mathscinet  zmath  isi  scopus  scopus
    38. Di H., Shang Ya., Zheng X., “Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms”, Discrete Contin. Dyn. Syst.-Ser. B, 21:3 (2016), 781–801  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    39. A. T. Asanova, “Periodic Solutions in the Plane of Systems of Second-Order Hyperbolic Equations”, Math. Notes, 101:1 (2017), 39–47  mathnet  crossref  crossref  mathscinet  isi  elib
    40. Di H., Shang Ya., “Existence, Nonexistence and Decay Estimate of Global Solutions For a Viscoelastic Wave Equation With Nonlinear Boundary Damping and Internal Source Terms”, Eur. J. Pure Appl Math., 10:4 (2017), 668–701  mathscinet  zmath  isi
    41. Glyzin S.D. Kolesov A.Yu. Preobrazhenskaia M.M., “Existence and Stability of Periodic Solutions of Quasi-Linear Korteweg - de Vries Equation”, V International Conference on Problems of Mathematical and Theoretical Physics and Mathematical Modelling, Journal of Physics Conference Series, 788, IOP Publishing Ltd, 2017, UNSP 012016  crossref  isi  scopus  scopus
    42. Luan W., Yang Z., “Global Existence and Bounds For Blow-Up Time in a Class of Nonlinear Pseudo-Parabolic Equations With a Memory Term”, Math. Meth. Appl. Sci., 42:8 (2019), 2597–2612  crossref  isi  scopus
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